Recurrence Relations

Abstract

A recurrence relation for a sequence {a _i }, which usually begins with a ₀ or a ₁, is a formula that defines a _n in terms of a ₀, a ₁, a ₂, a ₃,..., a _n−1, and n for all n greater than some particular integer k, with the terms a ₀, a ₁,..., a _k called initial conditions or boundary conditions. Together with the initial conditions, the recurrence relation provides a recursive definition for the elements of the sequence. This allows us to compute the unique value of a _n for each integer n such that n > k. Many examples follow. We are already familiar with several recurrence relations.